Can a minor request to only have her mother as her legal guardian even though the mother is still married to the minor's father?

Generally, no.  Your Mother can petition to be the sole guardian of you as a minor child under various circumstances and request the right to determine issues regarding you alone but you are not clear on the circumstances here.  Are your parents married and living together?  Are they separated but not divorced?  Are there divorce proceeding about to be filed or nothing in the immediate future? Is your Father not in the picture at all?

Should your Mother petition for sole guardianship and you are of an age the Court deems reasonable to have input, a Judge will listen to you may weigh your testimony and requests in the decision.  You are obviously not fond of the decisions made by your Father but you don't express why, and you have every right to privacy.  May I suggest that you seek some sort of counseling or confide in an adult you trust if there is something going on in your life that you need help with.  Maybe start with your school guidance counselor.  Good luck.